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This paper is concerned about maximum principles and radial symmetry forviscosity solutions of fully nonlinear partial differential equations. Weobtain the radial symmetry and monotonicity properties for nonnegativeviscosity solutions of begin{equation}F (D^2 u)+u^p=0 \quad \quad \mbox{in}\\mathbb R^n \label{abs}\end{equation} under the asymptotic decay rate$u=o(|x|^{-\frac{2}{p-1}})$ at infinity, where $p>1$ (Theorem 1, Corollary 1).As a consequence of our symmetry results, we obtain the nonexistence of anynontrivial and nonnegative solution when $F$ is the Pucci extremal operators(Corollary 2). Our symmetry and monotonicity results also apply toHamilton-Jacobi-Bellman or Isaccs equations. A new maximum principle forviscosity solutions to fully nonlinear elliptic equations is established(Theorem 2). As a result, different forms of maximum principles on bounded andunbounded domains are obtained. Radial symmetry, monotonicity and thecorresponding maximum principle for fully nonlinear elliptic equations in apunctured ball are shown (Theorem 3). We also investigate the radial symmetryfor viscosity solutions of fully nonlinear parabolic partial differentialequations (Theorem 4).

The Maximum Principles and Symmetry results for Viscosity Solutions of Fully Nonlinear Equations